Arbitrary Precision Research Articles

Data-driven approximations of topological insulator systems

A data-driven approach to calculating tight-binding models for discrete coupled-mode systems is presented. In particular, spectral and topological data are used to build an appropriate discrete model that accurately replicates these properties. This work is motivated by topological insulator systems that are often described by tight-binding models. The problem is formulated as the minimization of an appropriate residual (objective) function. Given bulk spectral data and a topological index (e.g., winding number), an appropriate discrete model is obtained to arbitrary precision. A nonlinear least squares method is used to determine the coefficients. The effectiveness of the scheme is highlighted against a Schrödinger equation with a periodic potential that can be described by the Su–Schrieffer–Heeger model.

Construction of Continuous Collective Energy Landscapes for Large Amplitude Nuclear Many-Body Problems.

Several protocols are proposed to build continuous energy surfaces of many-body quantum systems, regarding both energy and states. The standard variational principle is augmented with constraints on state overlap, ensuring arbitrary precision on continuity. As an illustration, the lowest energy and excited state paths relevant for the ^{240}Pu asymmetric fission are studied. The scission is clearly signed, with a neutron excess in the neck, the ultimate glue before its rupture. Our approach can potentially connect any couple of Hilbert space states, which opens up new horizons for various applications.

Collapse of Inelastic Hard Spheres in Dimension d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 2$$\\end{document}

We study the collapse of three inelastic particles in dimension d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 2$$\\end{document}. We obtain general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles. In the first case, we recover that the particles arrange in a nearly linear chain, already studied by Zhou and Kadanoff, and in the second case, we obtain that the particles arrange in an equilateral triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct an initial configuration leading to a nearly linear collapse, stable under perturbations, and such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision.

Algorithm 1048: A C++ Class for Robust Linear Barycentric Rational Interpolation

Barycentric rational interpolation is a recent interpolation method with several favourable properties. In this article, we present the BRI class, which features a new C++ class template that contains all variables and functions related to linear barycentric rational interpolation. While several methods exist to evaluate a barycentric rational interpolant, the class is designed to autonomously select the best method to use on a case-by-case basis, as it takes into account the latest results regarding the efficiency and numerical stability of barycentric rational interpolation [ 15 ]. Moreover, we describe a new technique that makes the code robust and less prone to overflow and underflow errors. In addition to the standard C++ data types, the BRI template variables can also be defined with arbitrary precision because the BRI class is compatible with the Multiple Precision Floating-Point Reliable (MPFR) library [ 14 ].

Squeezing the quantum noise of a gravitational-wave detector below the standard quantum limit

The Heisenberg uncertainty principle dictates that the position and momentum of an object cannot be simultaneously measured with arbitrary precision, giving rise to an apparent limitation known as the standard quantum limit (SQL). Gravitational-wave detectors use photons to continuously measure the positions of freely falling mirrors and so are affected by the SQL. We investigated the performance of the Laser Interferometer Gravitational-Wave Observatory (LIGO) after the experimental realization of frequency-dependent squeezing designed to surpass the SQL. For the LIGO Livingston detector, we found that the upgrade reduces quantum noise below the SQL by a maximum of three decibels between 35 and 75 hertz while achieving a broadband sensitivity improvement, increasing the overall detector sensitivity during astrophysical observations.

Deformable surface reconstruction via Riemannian metric preservation

Estimating the pose of an object from a monocular image is a fundamental inverse problem in computer vision. Due to its ill-posed nature, solving this problem requires incorporating deformation priors. In practice, many materials do not perceptibly shrink or extend when manipulated, constituting a reliable and well-known prior. Mathematically, this translates to the preservation of the Riemannian metric. Neural networks offer the perfect playground to solve the surface reconstruction problem as they can approximate surfaces with arbitrary precision and allow the computation of differential geometry quantities. This paper presents an approach for inferring continuous deformable surfaces from a sequence of images, which is benchmarked against several techniques and achieves state-of-the-art performance without the need for offline training. Being a method that performs per-frame optimization, our method can refine its estimates, contrary to those based on performing a single inference step. Despite enforcing differential geometry constraints at each update, our approach is the fastest of all the tested optimization-based methods.

Why are inner planets not inclined?

AbstractPoincaré’s work more than one century ago, or Laskar’s numerical simulations from the 1990’s on, have irrevocably impaired the long-held belief that the Solar System should be stable. But mathematical mechanisms explaining this instability have remained mysterious. In 1968, Arnold conjectured the existence of “Arnold diffusion” in celestial mechanics. We prove Arnold’s conjecture in the planetary spatial 4-body problem as well as in the corresponding hierarchical problem (where the bodies are increasingly separated), and show that this diffusion leads, on a long time interval, to some large-scale instability. Along the diffusive orbits, the mutual inclination of the two inner planets is close to $\pi /2$ π / 2 , which hints at why even marginal stability in planetary systems may exist only when inner planets are not inclined. More precisely, consider the normalised angular momentum of the second planet, obtained by rescaling the angular momentum by the square root of its semimajor axis and by an adequate mass factor (its direction and norm give the plane of revolution and the eccentricity of the second planet). It is a vector of the unit 3-ball. We show that any finite sequence in this ball may be realised, up to an arbitrary precision, as a sequence of values of the normalised angular momentum in the 4-body problem. For example, the second planet may flip from prograde nearly horizontal revolutions to retrograde ones. As a consequence of the proof, the non-recurrent set of any finite-order secular normal form accumulates on circular motions – a weak form of a celebrated conjecture of Herman.

Multilevel approximation of Gaussian random fields: Covariance compression, estimation, and spatial prediction

The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. We prove that a tapering strategy by thresholding applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. The locations of these nonzero matrix entries can be determined a priori. The tapered covariance or precision matrices may also be optimally diagonally preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. In addition, we propose and analyze novel compressive algorithms for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters p of the sample-wise approximation of the GRF in Sobolev scales.

Hamiltonian dynamics on digital quantum computers without discretization error

We introduce an algorithm to compute expectation values of time-evolved observables on digital quantum computers that requires only bounded average circuit depth to reach arbitrary precision, i.e. produces an unbiased estimator with finite average depth. This finite depth comes with an attenuation of the measured expectation value by a known amplitude, requiring more shots per circuit. The average gate count per circuit for simulation time t is O(t2μ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal{O}}({t}^{2}{\\mu }^{2})$$\\end{document} with μ the sum of the Hamiltonian coefficients, without dependence on precision, providing a significant improvement over previous algorithms. With shot noise, the average runtime is O(t2μ2ϵ−2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal{O}}({t}^{2}{\\mu }^{2}{\\epsilon }^{-2})$$\\end{document} to reach precision ϵ. The only dependence in the sum of the coefficients makes it particularly adapted to non-sparse Hamiltonians. The algorithm generalizes to time-dependent Hamiltonians, appearing for example in adiabatic state preparation. These properties make it particularly suitable for present-day relatively noisy hardware that supports only circuits with moderate depth.

Almost All Quantum Channels Are Diagonalizable

We prove the statement “The collection of all elements of[Formula: see text] which have only simple eigenvalues is dense in[Formula: see text]” for different sets [Formula: see text], including: all quantum channels, the unital channels, the positive trace-preserving maps, all Lindbladians (gksl-generators), and all time-dependent Markovian channels. Therefore any element from each of these sets can always be approximated by diagonalizable elements of the same set to arbitrary precision.

Transformations of probability distributions

Almost all of computer science is concerned with transformations of information in the form of strings. We initiate the study of a neglected transformation type, namely transformations between probability distributions. We begin by examining the deceivingly simple-looking case of Bernoulli distributions and procedures to transform them.A p-coin is a coin that, whenever tossed, lands heads up with probability p and tails up with probability 1−p. A neat trick due to von Neumann allows us to simulate a 1/2-coin with any p-coin for an arbitrary unknown p∈(0,1). We show how to apply this trick to simulate a q-coin for an arbitrary computable q∈(0,1). In contrast, it is impossible to simulate a q-coin with a noncomputable q.More generally, we are interested in what transformations between probability distributions are feasible. Is it possible to simulate a p/2-coin with a p-coin for unknown p? How about 2p or p2? We attempt to characterize the feasible transformations. For example, we show how to transform a p-coin into an f(p)-coin where any finite number of pairs (pi,f(pi)) of non-zero, non-one probabilities is prescribed, and that it is impossible to do the same for an infinite number of pairs. We also examine which probability distributions are feasible to approximate to arbitrary precision, showing that this is impossible for discontinuous ones but feasible for most but not all remaining ones.

Incompatible observables in classical physics: A closer look at measurement in Hamiltonian mechanics.

Quantum theory famously entails the existence of incompatible measurements, pairs of system observables which cannot be simultaneously measured to arbitrary precision. Incompatibility is widely regarded to be a uniquely quantum phenomenon, linked to failure to commute of quantum operators. Even in the face of deep parallels between quantum commutators and classical Poisson brackets, no connection has been established between the Poisson algebra and any intrinsic limitations to classical measurement. Here I examine measurement in classical Hamiltonian physics as a process involving the joint evolution of an object-system and a finite-temperature measuring apparatus. Instead of the ideal measurement capable of extracting information without disturbing the system, I find a Heisenberg-like precision-disturbance relation: Measuring an observable leaves all Poisson-commuting observables undisturbed but inevitably disturbs all non-Poisson-commuting observables. In this classical uncertainty relation the role of h-bar is played by an apparatus-specific quantity, q-bar. While this is not a universal constant, the analysis suggests that q-bar takes a finite positive value for any apparatus that can be built. (Specifically: q-bar vanishes in the model only in the unreachable limit of zero absolute temperature.) I show that a classical version of Ozawa's model of quantum measurement [Ozawa, Phys. Rev. Lett. 60, 385 (1988)0031-900710.1103/PhysRevLett.60.385], originally proposed as a means to violate Heisenberg's relation, does not violate the classical relation. If this result were to generalize to all models of measurement, then incompatibility would prove to be a feature not only of quantum, but of classical physics too. Put differently: The approach presented here points the way to studying the (Bayesian) epistemology of classical physics, which was until now assumed to be trivial. It now seems possible that it is nontrivial and bears a resemblance to the quantum formalism. The present findings may be of interest to researchers working on foundations of quantum mechanics, particularly for ψ-epistemic interpretations. More practically, there may be applications in the fields of precision measurement, nanoengineering, and molecular machines.

Conditional quantum thermometry—enhancing precision by measuring less

Taking accurate measurements of the temperature of quantum systems is a challenging task. The mathematical peculiarities of quantum information make it virtually impossible to measure with infinite precision. In the present paper, we introduce a generalize thermal state, which is conditioned on the pointer states of the available measurement apparatus. We show that this conditional thermal state outperforms the Gibbs state in quantum thermometry. The origin for the enhanced precision can be sought in its asymmetry quantified by the Wigner–Yanase–Dyson skew information. This additional resource is further clarified in a fully resource-theoretic analysis, and we show that there is a Gibbs-preserving map to convert a target state into the conditional thermal state. We relate the quantum J-divergence between the conditional thermal state and the same target state to quantum heat.

PyChelator: a Python-based Colab and web application for metal chelator calculations

BackgroundMetal ions play vital roles in regulating various biological systems, making it essential to control the concentration of free metal ions in solutions during experimental procedures. Several software applications exist for estimating the concentration of free metals in the presence of chelators, with MaxChelator being the easily accessible choice in this domain. This work aimed at developing a Python version of the software with arbitrary precision calculations, extensive new features, and a user-friendly interface to calculate the free metal ions.ResultsWe introduce the open-source PyChelator web application and the Python-based Google Colaboratory notebook, PyChelator Colab. Key features aim to improve the user experience of metal chelator calculations including input in smaller units, selection among stability constants, input of user-defined constants, and convenient download of all results in Excel format. These features were implemented in Python language by employing Google Colab, facilitating the incorporation of the calculator into other Python-based pipelines and inviting the contributions from the community of Python-using scientists for further enhancements. Arbitrary-precision arithmetic was employed by using the built-in Decimal module to obtain the most accurate results and to avoid rounding errors. No notable differences were observed compared to the results obtained from the PyChelator web application. However, comparison of different sources of stability constants showed substantial differences among them.ConclusionsPyChelator is a user-friendly metal and chelator calculator that provides a platform for further development. It is provided as an interactive web application, freely available for use at https://amrutelab.github.io/PyChelator, and as a Python-based Google Colaboratory notebook at https://colab.research.google.com/github/AmruteLab/PyChelator/blob/main/PyChelator_Colab.ipynb.

On the Expansion of Resolvents and the Integrated Density of States for Poisson Distributed Schrödinger Operators

We consider a Schrödinger operator with random potential distributed according to a Poisson process. We show that under a uniform moment bound expectations of matrix elements of the resolvent as well as the integrated density of states can be approximated to arbitrary precision in powers of the coupling constant. The expansion coefficients are given in terms of expectations obtained by Neumann expanding the potential around the free Laplacian. Our results are valid for arbitrary strength of the disorder parameter, including the small disorder regime.

Full asymptotic expansion for orbit-summable quadrant walks and discrete polyharmonic functions

Enumeration of walks with small steps in the quadrant has been a topic of great interest in combinatorics over the last few years. In this article, it is shown how to compute exact asymptotics of the number of such walks with fixed start- and endpoints for orbit-summable models with finite group, up to arbitrary precision. The resulting representation greatly resembles one conjectured for walks starting from the origin in 2020 by Chapon, Fusy and Raschel, differing only in terms appearing due to the periodicity of the model. We will see that the dependency on start- and endpoint is given by discrete polyharmonic functions, which are solutions of △nv=0 for a discretization △ of a Laplace–Beltrami operator. They can be decomposed into a sum of products of lower order polyharmonic functions of either the start- or the endpoint only, which leads to a partial extension of a theorem by Denisov and Wachtel.

Designing optimal protocols in Bayesian quantum parameter estimation with higher-order operations

Using quantum systems as sensors or probes has been shown to greatly improve the precision of parameter estimation by exploiting unique quantum features such as entanglement. A major task in quantum sensing is to design the optimal protocol, i.e., the most precise one. It has been solved for some specific instances of the problem, but in general even numerical methods are not known. Here, we focus on the single-shot Bayesian setting, where the goal is to find the optimal initial state of the probe (which can be entangled with an auxiliary system), the optimal measurement, and the optimal estimator function. We leverage the formalism of higher-order operations to develop a method based on semidefinite programming that finds a protocol that is close to the optimal one with arbitrary precision. Crucially, our method is not restricted to any specific quantum evolution, cost function, or prior distribution, and thus can be applied to any estimation problem. Moreover, it can be applied to both single-parameter or multiparameter estimation tasks. We demonstrate our method with three examples, consisting of unitary phase estimation, thermometry in a bosonic bath, and multiparameter estimation of an SU(2) transformation. Exploiting our methods, we extend several results from the literature. For example, in the thermometry case, we find the optimal protocol at any finite time and quantify the usefulness of entanglement. Published by the American Physical Society 2024

A Sheaf-Theoretic Construction of Shape Space

AbstractWe present a sheaf-theoretic construction of shape space—the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transforms (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to “glue” PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision.

The Emergence of Consciousness in a Physical Universe

Consciousness appears so mysterious and hard to formulate within physical sciences because the present day scientific thinking excludes an element of reality and a general mechanics of its processing from its consideration. The primary missing element is the reality of information in the physical universe as an intrinsic causal correlate of observable physical states. Moreover, there exists a general formalism of information processing that is universally applicable to the processing resulting from each physical interaction. As shown, the formalism further enables a general mechanism to construct arbitrary structured and abstract semantics or object description in modular hierarchy as well as a powerful mechanism of population coding to represent arbitrary precision and variation in object description resolving the combinatorial problem. Here, a semantic value, or simply semantics, is equivalent (\(\equiv\)) to the content of information of causal correlation, and treated as a value to enable its formal processing. The primary motive here is to lay down a formal account of information (semantic) processing that leads to bridging the conceptual gap between the objectively observable elements in nature and the subjective consciousness. It is shown that the qualities we associate with consciousness are causally correlated semantics of relation that a represented agency holds with other objects within a dynamically evolving semantic structure, where the state of the population of physical systems (neurons) correlating with the structure holds causal powers to effect appropriate behavior. Since the information (semantic value) arises from natural causal dependence, the correlation based consciousness forms an undeniable reality of existence. It is derived here how a semantic value equivalent to ‘a self as an observer of objects and controller of actions’ is constructed. If the semantic components of a conscious experience, such as the self, the objects of experience, and the relation of experience attributing the self as the owner or experiencer, causally correlate with a system’s state having causal influence in action, then it suffices to bridge the gap between the objective reality and the subjective consciousness. That is, the semantic value corresponding to the thoughts and senses is the reality of nature the semantics of self relates to as the owner. Moreover, the semantics of ‘self as an observer and controller of action’ is itself shown to form a part of observed objects giving rise to self awareness.

Effect of Phase Shifting on Real-Time Detection and Classification of Power Quality Disturbances

Power quality improvement and Power quality disturbance (PQD) detection are two significant concerns that must be addressed to ensure an efficient power distribution within the utility grid. When the process to analyze PQD is migrated to real-time platforms, the possible occurrence of a phase mismatch can affect the algorithm’s accuracy; this paper evaluates phase shifting as an additional stage in signal acquisition for detecting and classifying eight types of single power quality disturbances. According to their mathematical models, a set of disturbances was generated using an arbitrary waveform generator BK Precision 4064. The acquisition, detection, and classification stages were embedded into a BeagleBone Black. The detection stage was performed using multiresolution analysis. The feature vectors of the acquired signals were obtained from the combination of Shannon entropy and log-energy entropy. For classification purposes, four types of classifiers were trained: multilayer perceptron, K-nearest neighbors, probabilistic neural network, and decision tree. The results show that incorporating a phase-shifting stage as a preprocessing stage significantly improves the classification accuracy in all cases.